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Properties

Label 13.1.101439097511492601.1
Degree $13$
Signature $[1, 6]$
Discriminant $1.014\times 10^{17}$
Root discriminant \(20.33\)
Ramified primes $3,7,17$
Class number $1$
Class group trivial
Galois group $C_{13}:C_6$ (as 13T5)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^13 - 4*x^12 + 10*x^11 - 25*x^10 + 52*x^9 - 75*x^8 + 90*x^7 - 113*x^6 + 114*x^5 - 65*x^4 + 10*x^3 + 37*x^2 - 19*x - 10)
 
Copy content gp:K = bnfinit(y^13 - 4*y^12 + 10*y^11 - 25*y^10 + 52*y^9 - 75*y^8 + 90*y^7 - 113*y^6 + 114*y^5 - 65*y^4 + 10*y^3 + 37*y^2 - 19*y - 10, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 4*x^12 + 10*x^11 - 25*x^10 + 52*x^9 - 75*x^8 + 90*x^7 - 113*x^6 + 114*x^5 - 65*x^4 + 10*x^3 + 37*x^2 - 19*x - 10);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^13 - 4*x^12 + 10*x^11 - 25*x^10 + 52*x^9 - 75*x^8 + 90*x^7 - 113*x^6 + 114*x^5 - 65*x^4 + 10*x^3 + 37*x^2 - 19*x - 10)
 

\( x^{13} - 4 x^{12} + 10 x^{11} - 25 x^{10} + 52 x^{9} - 75 x^{8} + 90 x^{7} - 113 x^{6} + 114 x^{5} + \cdots - 10 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $13$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[1, 6]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(101439097511492601\) \(\medspace = 3^{6}\cdot 7^{8}\cdot 17^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(20.33\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}7^{2/3}17^{1/2}\approx 26.132669826287227$
Ramified primes:   \(3\), \(7\), \(17\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{11}+\frac{2}{9}a^{8}-\frac{1}{3}a^{7}+\frac{1}{9}a^{6}-\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{3}a^{3}+\frac{4}{9}a-\frac{1}{9}$, $\frac{1}{9137277}a^{12}-\frac{360289}{9137277}a^{11}+\frac{59423}{1015253}a^{10}+\frac{911597}{9137277}a^{9}+\frac{1650913}{9137277}a^{8}-\frac{3067700}{9137277}a^{7}+\frac{479381}{1015253}a^{6}+\frac{492733}{1015253}a^{5}-\frac{920389}{9137277}a^{4}-\frac{199771}{3045759}a^{3}+\frac{3537187}{9137277}a^{2}-\frac{1120514}{9137277}a-\frac{3862208}{9137277}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $6$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{432005}{9137277}a^{12}-\frac{419258}{3045759}a^{11}+\frac{361010}{1015253}a^{10}-\frac{8268233}{9137277}a^{9}+\frac{5288383}{3045759}a^{8}-\frac{21540010}{9137277}a^{7}+\frac{28061420}{9137277}a^{6}-\frac{34639075}{9137277}a^{5}+\frac{3386368}{1015253}a^{4}-\frac{6581108}{3045759}a^{3}+\frac{3859322}{9137277}a^{2}+\frac{11071348}{9137277}a-\frac{1012007}{3045759}$, $\frac{4249}{1015253}a^{12}+\frac{186814}{9137277}a^{11}-\frac{239066}{3045759}a^{10}+\frac{185458}{1015253}a^{9}-\frac{5004784}{9137277}a^{8}+\frac{1191220}{1015253}a^{7}-\frac{13883930}{9137277}a^{6}+\frac{15867817}{9137277}a^{5}-\frac{22140311}{9137277}a^{4}+\frac{6427532}{3045759}a^{3}-\frac{2924053}{3045759}a^{2}+\frac{3238003}{9137277}a+\frac{7356071}{9137277}$, $\frac{1254055}{9137277}a^{12}-\frac{4179305}{9137277}a^{11}+\frac{1155318}{1015253}a^{10}-\frac{26773297}{9137277}a^{9}+\frac{53104928}{9137277}a^{8}-\frac{72932165}{9137277}a^{7}+\frac{91673633}{9137277}a^{6}-\frac{116220541}{9137277}a^{5}+\frac{103928545}{9137277}a^{4}-\frac{5750142}{1015253}a^{3}+\frac{8001757}{9137277}a^{2}+\frac{12220271}{3045759}a+\frac{4608287}{9137277}$, $\frac{48957}{1015253}a^{12}-\frac{1027571}{9137277}a^{11}+\frac{186682}{1015253}a^{10}-\frac{452298}{1015253}a^{9}+\frac{5260229}{9137277}a^{8}+\frac{931664}{3045759}a^{7}-\frac{11898608}{9137277}a^{6}+\frac{15621751}{9137277}a^{5}-\frac{30111968}{9137277}a^{4}+\frac{16979525}{3045759}a^{3}-\frac{5701263}{1015253}a^{2}+\frac{30895396}{9137277}a+\frac{18887819}{9137277}$, $\frac{384920}{9137277}a^{12}-\frac{642613}{3045759}a^{11}+\frac{466323}{1015253}a^{10}-\frac{9977150}{9137277}a^{9}+\frac{7526420}{3045759}a^{8}-\frac{30097603}{9137277}a^{7}+\frac{29205737}{9137277}a^{6}-\frac{44767852}{9137277}a^{5}+\frac{5917182}{1015253}a^{4}-\frac{1550704}{1015253}a^{3}-\frac{5534212}{9137277}a^{2}-\frac{12530432}{9137277}a-\frac{2383571}{3045759}$, $\frac{293817}{1015253}a^{12}-\frac{7730287}{9137277}a^{11}+\frac{5922503}{3045759}a^{10}-\frac{15219916}{3045759}a^{9}+\frac{84988264}{9137277}a^{8}-\frac{10950030}{1015253}a^{7}+\frac{115908875}{9137277}a^{6}-\frac{152588842}{9137277}a^{5}+\frac{112808570}{9137277}a^{4}-\frac{2867401}{1015253}a^{3}-\frac{2426743}{1015253}a^{2}+\frac{90970382}{9137277}a+\frac{33301231}{9137277}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4878.48838792 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 4878.48838792 \cdot 1}{2\cdot\sqrt{101439097511492601}}\cr\approx \mathstrut & 0.942457548881 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^13 - 4*x^12 + 10*x^11 - 25*x^10 + 52*x^9 - 75*x^8 + 90*x^7 - 113*x^6 + 114*x^5 - 65*x^4 + 10*x^3 + 37*x^2 - 19*x - 10) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^13 - 4*x^12 + 10*x^11 - 25*x^10 + 52*x^9 - 75*x^8 + 90*x^7 - 113*x^6 + 114*x^5 - 65*x^4 + 10*x^3 + 37*x^2 - 19*x - 10, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 4*x^12 + 10*x^11 - 25*x^10 + 52*x^9 - 75*x^8 + 90*x^7 - 113*x^6 + 114*x^5 - 65*x^4 + 10*x^3 + 37*x^2 - 19*x - 10); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 4*x^12 + 10*x^11 - 25*x^10 + 52*x^9 - 75*x^8 + 90*x^7 - 113*x^6 + 114*x^5 - 65*x^4 + 10*x^3 + 37*x^2 - 19*x - 10); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{13}:C_6$ (as 13T5):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 78
The 8 conjugacy class representatives for $C_{13}:C_6$
Character table for $C_{13}:C_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 26 sibling: data not computed
Degree 39 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ R ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.13.0.1}{13} }$ R ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.13.0.1}{13} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.13.0.1}{13} }$ ${\href{/padicField/43.13.0.1}{13} }$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.3.2.3a1.1$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
3.3.2.3a1.1$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
7.2.3.4a1.2$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$
7.2.3.4a1.2$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$$[\ ]$$
17.3.2.3a1.1$x^{6} + 2 x^{4} + 28 x^{3} + x^{2} + 45 x + 196$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
17.3.2.3a1.1$x^{6} + 2 x^{4} + 28 x^{3} + x^{2} + 45 x + 196$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)